# Thread: Learning Calculus - Finding turning points.

1. ## Learning Calculus - Finding turning points.

Hi everyone,

I'm trying to teach myself calculus and have come across this problem.

I have a function of x

f(x) = x^3 + 5x^2 - 8x - 12

so I set this to zero by

f'(x) = 3x^2 + 10x - 8 = 0

then the book I'm using says that I can find the local values of x for min and max and these should be x1 = 0.667 and x2 = -4.000 but it doesn't say how these figures are reached.

Could someone guide me in the right direction please.

Andy

2. ## Re: Learning Calculus Originally Posted by Rastanking Hi everyone,

I'm trying to teach myself calculus and have come across this problem.

I have a function of x

f(x) = x^3 + 5x^2 - 8x - 12

so I set this to zero by

f'(x) = 3x^2 + 10x - 8 = 0

then the book I'm using says that I can find the local values of x for min and max and these should be x1 = 0.667 and x2 = -4.000 but it doesn't say how these figures are reached.

Could someone guide me in the right direction please.

Andy
By solving $\displaystyle f'(x) = 0$ - in this case using your favourite method for solving quadratic equations

3. ## Re: Learning Calculus Originally Posted by Rastanking Hi everyone,

I'm trying to teach myself calculus and have come across this problem.

I have a function of x

f(x) = x^3 + 5x^2 - 8x - 12

so I set this to zero by

f'(x) = 3x^2 + 10x - 8 = 0

then the book I'm using says that I can find the local values of x for min and max and these should be x1 = 0.667 and x2 = -4.000 but it doesn't say how these figures are reached.

Could someone guide me in the right direction please.

Andy
When you have a quadratic equation of the form

$\displaystyle ax^2+bx+c=0$

The "x" solutions are simple to see if you can write your sum of 3 terms as a product by factoring to

$\displaystyle (3x+?)(x+?)=0$

We take the factors of -8 that sum to 10 when multiplying out the brackets.
This is (3)4+1(-2) and so

$\displaystyle 3x^2+10x-8=0\Rightarrow\ (3x-2)(x+4)=0$

Since 0(anything)=0

then those factors are giving you the 2 solutions

$\displaystyle 3x-2=0$

$\displaystyle x+4=0$

Alternatively,

for $\displaystyle ax^2+bx+c=0$

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

4. ## Re: Learning Calculus

Thanks for the quick reply, it has been a great help. However the answers in the book are 0.667 and -4.000 but I end up with -1.33 and -2.

I'm either losing the plot entirely or having finger trouble, could someone possible check my maths please.

5. ## Re: Learning Calculus

You should show how you are getting those answers.
Are you factoring or using the "quadratic formula" ?

6. ## Re: Learning Calculus

I'm using the quadratic formula and ending up with -1.33 and -2

7. ## Re: Learning Calculus

$\displaystyle f'(x)=0\Rightarrow\ 3x^2+10x-8=0$

$\displaystyle \Rightarrow\ a=3,\;\;b=10,\;\;c=-8$

$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-10\pm\sqrt{100-4(3)(-8)}}{6}$

$\displaystyle x=\frac{-10\pm\sqrt{196}}{6}$

$\displaystyle x_1=\frac{\sqrt{196}-10}{6}$

$\displaystyle x_2=\frac{-10-\sqrt{196}}{6}$

One of these is positive and the other is negative.

Your error was to take c=8 instead of minus 8.

$\displaystyle ax^2+bx+c=(3)x^2+(10)x+(-8)$

8. ## Re: Learning Calculus

Many thanks again for your help, very much appreciated.

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